Stability of the NLS equation with viscosity effect. (English) Zbl 1216.35141

Karjanto, N.; Tiong, K. M.

J. Appl. Math. 2011, Article ID 863161, 11 p. (2011).

Summary: A nonlinear Schrödinger (NLS) equation with an effect of viscosity is derived from a Korteweg-de Vries (KdV) equation modified with viscosity using the method of multiple time scale. It is well known that the plane-wave solution of the NLS equation exhibits modulational instability phenomenon. In this paper, the modulational instability of the plane-wave solution of the NLS equation modified with viscosity is investigated. The corresponding modulational dispersion relation is expressed as a quadratic equation with complex-valued coefficients. By restricting the modulational wavenumber into the case of narrow-banded spectra, it is observed that a type of dissipation, in this case the effect of viscosity, stabilizes the modulational instability, as confirmed by earlier findings.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B35	Stability in context of PDEs

Keywords:

nonlinear Schrödinger equation; modulational instability

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